Macro and micro view on steady states in state space

Acta Univ. Sapientiae, Informatica, 2, 1 (2010) 90–98 Macro and micro view on steady states in state space Branislav Sobota Dept. of Computers and Informatics FEEI, Technical University of Koˇsice Letn´a 9, 04200 Koˇsice, Slovakia email: Branislav.Sobota@tuke.sk Milan Guzan Dept. of Theoretical Electrotechnics and Electrical Measurement FEEI, Technical University of Koˇsice Letn´a 9, 04200 Koˇsice, Slovakia email: Milan.Guzan@tuke.sk Abstract. This paper describes visualization of chaotic attractor and elements of the singularities in 3D space. 3D view of these effects enables to create a demonstrative projection about relations of chaos generated by physical circuit, the Chua’s circuit. Via macro views on chaotic at- tractor is obtained not only visual space illustration of representative point motion in state space, but also its relation to planes of singularity elements. Our created program enables view on chaotic attractor both in 2D and 3D space together with plane objects visualization – elements of singularities. 1 Introduction The visualization is good idea to show imagines, ideas, design, construction, realization or effects. It is also one way of verification before realization our goals. Computer based visualization brings utilization of physical, or simu- lated electric parameters course graphical interpretation in non-linear circuit theory together with other fields. From beginning it was used 2D visualization with possibility of color utilization to be more illustrative or to explain actions proceeding in non-linear circuits [1, 9, 13, 15]. High-performance or parallel computers enable to take advantage of 3D state space axonometry [14]. Ac- tual available solutions provide high performance visualization suitable for 3D Computing Classification System 1998: B.7.2 Mathematics Subject Classification 2010: 94C99, 68U05 Key words and phrases: chaos, Chua’s circuit, singularity, trajectory, 3D visualization 90 arXiv:1003.1401v1 [cs.GR] 6 Mar 2010 Macro and micro view on steady states in state space 91 interactive presentation of processes and effects with support for over million saturated colors in hi-resolution mode and for use in all graphics-intensive ap- plications. This paper describes actual possibilities of PC for visualization of steady states of chaos generating circuit. 2 Methods used for trajectory visualization In last 24 years there was intensive interest of scientific community to analyse and applied Chua’s circuit generating chaos. Presentation of trajectories needs to solve system (1) describing physical Chua’s circuit. C1(du1/dt) = G(u2 −u1) −g(u1) −I = Q1 C2(du2/dt) = G(u1 −u2) + i = Q2 (1) L(di/dt) = −u2 −ρi = Q3 where g(u1) = m2u1 + 1/2(m1 −m0)(|u1 −BP| −|u1 + BP|) + +1/2(m2 −m1)(|u1 −B0| −|u1 + B0|) (2) Next we consider control pulse I = 0, the resistance of the inductance ρ = 0. For parameters (3) in [5] there were found chaotic attractors showed in Fig. 1 in Monge projection. C1 = 1/9, C2 = 1, L = 0.142857, G = 0.7, m0 = −0.8, m1 = −0.5, m2 = 5, Bp = 1, B0 = 14 (3) Computer program was designed in C language by author of [2] and used for clarifying of place in state space where chaos originates [3]. It is only short segment of intersection two surfaces related to circuit singularities P+ and 0, or 0 and P−. In despite of explaining with help of tables and 2D presen- tation was definite, 3D visualization provides faster and lighter illustration of actions which proceed in specific non-linear circuit. Therefore 3D visualization is valued as from scientific as from edifying point of view [4]. 3 Visualization in 3-dimensional space Chaos visualizing system was designed for visualization Chua’s attractor in 3D space in real time and it is based on visualizing kernel developed on DCI 92 B. Sobota, M. Guzan Figure 1: Monge projection of chaotic attractor system (1), for parameters (3) to plane: a) i −u1, b) i −u2 FEEI TU Koˇsice [6]. An application is implemented in C++ language using OpenGL graphics library. The application can work with I-V characteristics, Chua’s attractor trajectory, or limit cycle and it can visualize elements of the singularity planes. Additionally this visualization depicts representative point movement and it creates chaotic attractor using two basic modes (continuous and sequential). This application can be used not only for concrete Chua’s circuit. It is usable also for Chua’s circuit like structures analyzed in [7, 8]. Chaos visualizing system provides three basic visualizing modes: continuous mode, sequential mode and I-V characteristics visualization mode. System al- lows using four projection types (3D u1−i−u2 projection (see Fig. 5), 2D i−u2 projection, 2D i−u1 projection and 2D u2−u1 projection) for better-examined circuit understanding. Settable basic visualizing parameters for chaotic attrac- tor visualization are: drawing speed, points omission, chaotic attractor point size, comet length and attractor colour. The combination of these parameters defines fina

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